Study Notes: Stochastic Calculus for Finance II - Continuous-Time Models

General Principles of Continuous-Time Finance

• Continuous-Time Modeling: Unlike discrete-time models, continuous-time models assume that trading can occur at any instant. This framework uses stochastic calculus, primarily Itô calculus, to model asset prices.
• Asset Price Dynamics: Asset prices are typically modeled using Geometric Brownian Motion (GBM), represented by the stochastic differential equation (SDE): $dS(t) = \alpha S(t) dt + \sigma S(t) dW(t)$, where $\alpha$ is the mean return and $\sigma$ is the volatility.

Stochastic Calculus and Itô's Integral

• Brownian Motion ($W(t)$): A continuous stochastic process where $W(0) = 0$, paths are continuous, and increments are independent and normally distributed: $W(t) - W(s) \sim N(0, t-s)$.
• Itô Integral: Defined as $\int_0^T \Delta(t) dW(t)$. Unlike Riemann integrals, Itô integrals are used when integrating with respect to Brownian motion. A key property is the Martingale property: $E[\int_0^t \Delta(u) dW(u)] = 0$.
• Itô-Doeblin Formula: The fundamental tool for finding the differential of a function of a stochastic process. For $X(t)$ following $dX(t) = a(t) dt + b(t) dW(t)$, the differential of $f(t, X(t))$ is:
  $df(t, X(t)) = [f_t(t, X(t)) + a(t) f_x(t, X(t)) + \frac{1}{2} b^2(t) f_{xx}(t, X(t))] dt + b(t) f_x(t, X(t)) dW(t)$

Risk-Neutral Pricing and Martingales

• The Money Market Account: Value grows at the risk-free rate $r(t)$: $dM(t) = r(t) M(t) dt$. The discount factor is $D(t) = e^{-\int_0^t r(s) ds}$.
• Risk-Neutral Measure ($\tilde{P}$): A probability measure under which the discounted stock price $D(t)S(t)$ is a martingale. This is also called the Equivalent Martingale Measure (EMM).
• Girsanov’s Theorem: Provides the mechanism to change the measure from the actual probability $P$ to the risk-neutral probability $\tilde{P}$ by changing the drift of the Brownian motion.
• Market Completeness: A market is complete if every contingent claim (derivative) can be replicated by a trading strategy involving the underlying assets.

Black-Scholes-Merton Framework

• The Black-Scholes Equation: A partial differential equation (PDE) that the price of a derivative $f(t, s)$ must satisfy:
  $f_t + rsf_s + \frac{1}{2} \sigma^2 s^2 f_{ss} - rf = 0$
• The Black-Scholes Formula for Call Options:
  $C(t, s) = s N(d_1) - K e^{-r(T-t)} N(d_2)$
  where:
  $d_1 = \frac{\ln(s/K) + (r + \sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}$
  $d_2 = d_1 - \sigma \sqrt{T-t}$
• Delta Hedging: To replicate a derivative, one must hold $\Delta = \frac{\partial f}{\partial s}$ units of the underlying stock at any time $t$. This makes the portfolio insensitive to small changes in the stock price.

Exotic Options and Interest Rate Models

• Exotic Options: Beyond standard calls and puts, these include barrier options (active or inactive based on price thresholds), Asian options (payoff depends on average price), and lookback options.
• Term Structure of Interest Rates: Models like Vasicek or Cox-Ingersoll-Ross (CIR) describe the evolution of the short rate $r(t)$.
  • Vasicek Model: $dr(t) = (a - b r(t)) dt + \sigma dW(t)$ (Mean-reverting).
  • CIR Model: $dr(t) = (a - b r(t)) dt + \sigma \sqrt{r(t)} dW(t)$ (Ensures rates remain positive).

Feynman-Kac Theorem

• Theorem Description: Establishes a link between parabolic PDEs and stochastic processes. It states that the solution to a specific PDE can be expressed as an expectation of a stochastic process under the risk-neutral measure:
  $f(t, x) = \tilde{E} [e^{-\int_t^T r(u) du} h(X(T)) | X(t) = x]$

Change of Num raire

• Num raire: An asset used to denominate the value of other assets. Moving from the money market account to another asset (e.g., the stock itself or a zero-coupon bond) as the num raire can simplify the pricing of complex derivatives.
• Fundamental Theorem of Asset Pricing: In an arbitrage-free market, the price of an asset divided by the num raire is a martingale under a measure associated with that num raire.